How to solve Fokker-Planck equation treating mixed eigenvalue spectrum?

An analogy of the Fokker-Planck equation (FPE) with the Schrodinger equation allows us to use quantum mechanics technique to find the analytical solution of the FPE in a number of cases. However, previous studies have been limited to the Schrodinger potential with a discrete eigenvalue spectrum. Here, we will show how this approach can be also applied to a mixed eigenvalue spectrum with bounded and free states. We solve the FPE with boundaries located at x = +/- L/2 and take the limit L -> infinity, considering the examples with constant Schrodinger potential and with Poschl-Teller potential. An oversimplified approach was proposed earlier by M.T. Araujo and E. Drigo Filho. A detailed investigation of the two examples shows that the correct solution, obtained in this paper, is consistent with the expected Fokker-Planck dynamics.